Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in 3-space

2008 ◽  
pp. 73-106
Author(s):  
Ernst Binz ◽  
Sonja Pods
Keyword(s):  
Vector Fields ◽  
Chern Classes ◽  
Homotopy Classes ◽  
Isomorphism Classes

Immersions with non-zero normal vector fields

1992 ◽  
Vol 112 (2) ◽  
pp. 281-285 ◽  
Author(s):  
Bang-He Li ◽  
Gui-Song Li
Keyword(s):  
Vector Fields ◽  
Normal Vector ◽  
Homotopy Classes ◽  
Natural Action ◽  
Singularity Method ◽  
One To One ◽  
Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

Derivation Hom-Lie 2-algebras and non-abelian extensions of regular Hom-Lie algebras

2018 ◽  
Vol 17 (05) ◽  
pp. 1850081 ◽  
Author(s):  
Lina Song ◽  
Rong Tang
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Homotopy Classes ◽  
Abelian Extensions ◽  
Isomorphism Classes ◽  
One To One

In this paper, we introduce the notion of a derivation of a regular Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of regular Hom-Lie algebras. We show that isomorphism classes of diagonal non-abelian extensions of a regular Hom-Lie algebra [Formula: see text] by a regular Hom-Lie algebra [Formula: see text] are in one-to-one correspondence with homotopy classes of morphisms from [Formula: see text] to the derivation Hom-Lie 2-algebra [Formula: see text].

scholarly journals On Homotopy Invariants of Combings of Three-manifolds

2015 ◽  
Vol 67 (1) ◽  
pp. 152-183 ◽  
Author(s):  
Christine Lescop
Keyword(s):  
Vector Fields ◽  
Euler Class ◽  
Integral Invariant ◽  
Alternative Definition ◽  
Rational Homology ◽  
Homotopy Classes ◽  
Torsion Element ◽  
3 Dimensional ◽  
Linking Number ◽  
Homotopy Invariants

AbstractCombings of compact, oriented, 3-dimensional manifoldsMare homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying Spinc-structure. A combing is called torsion if this Euler class is a torsion element of H2(M; Z). Gompf introduced a Q-valued invariant θGof torsion combings on closed 3-manifolds, and he showed that θGdistinguishes all torsion combings with the same Spinc-structure. We give an alternative definition for θGand we express its variation as a linking number. We define a similar invariantp1of combings for manifolds bounded by S2. We relate p1 to the Θ-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula Θ = ¼P1+ 6λ, where λ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

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